Oxford

I will describe some recent efforts to recreate the miraculous properties of perverse sheaves on complex analytic spaces in the setting of real stratified spaces.

We use Non-Reductive GIT to construct compactifications of Hilbert schemes of complete intersections. We then study ample line bundles on these compactifications in order to construct moduli spaces of complete intersections for certain degree types.

''Positive geometries'' are a class of semi-algebraic domains which admit a unique ''canonical form'': a logarithmic form whose residues match the boundary structure of the domain. The study of such geometries is motivated by recent progress in particle physics, where the corresponding canonical forms are interpreted as the integrands of scattering amplitudes. We recast these concepts in the language of mixed Hodge theory, and identify ''genus zero pairs'' of complex algebraic varieties as a natural and general framework for the study of positive geometries and their canonical forms. In this framework, we prove some basic properties of canonical forms which have previously been proved or conjectured in the literature. We give many examples and study in detail the case of arrangements of hyperplanes and convex polytopes.

[arXiv:2501.03202]

Hoheisel's theorem states that there is some $\delta> 0$ and some $x_0>0$ such that for all $x > x_0$ the interval $[x,x+x^{1-\delta}]$ contains prime numbers. Classically this is proved using the Riemann zeta function and results about its zeros such as the zero-free region and zero density estimates. In this talk I will describe a new elementary proof of Hoheisel's theorem. This is joint work with Kaisa Matomäki (Turku) and Joni Teräväinen (Cambridge). Instead of the zeta function, our approach is based on sieve methods and ideas coming from additive combinatorics, in particular, the transference principle. The method also gives an L-function free proof of Linnik's theorem on the least prime in arithmetic progressions.

A well-known problem in algebraic geometry is to construct smooth projective Calabi-Yau varieties $Y$. In the smoothing approach, we construct first a degenerate (reducible) Calabi-Yau scheme $V$ by gluing pieces. Then we aim to find a family $f\colon X \to C$ with special fiber $X_0 = f^{-1}(0) \cong V$ and smooth general fiber $X_t = f^{-1}(t)$. In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber $V$. This is achieved via the logarithmic Bogomolov-Tian-Todorov theorem as well as its variant for pairs of a log Calabi-Yau space $f_0\colon X_0 \to S_0$ and a line bundle $\mathcal{L}_0$ on $X_0$.

Derived algebraic geometry allows us to study formal moduli problems via their tangent Lie algebras. After briefly reviewing this general paradigm, I will explain how it sheds light on deformations of Calabi-Yau varieties. 
In joint work with Taelman, we prove a mixed characteristic analogue of the Bogomolov–Tian–Todorov theorem, which asserts that Calabi-Yau varieties in characteristic $0$ are unobstructed. Moreover, we show that ordinary Calabi–Yau varieties in characteristic $p$ admit canonical (and algebraisable) lifts to characteristic $0$, generalising results of Serre-Tate for abelian varieties and Deligne-Nygaard for K3 surfaces. 
If time permits, I will conclude by discussing some intriguing questions related to our canonical lifts.  
 

Gauging is a systematic way to construct a model with non-invertible symmetry from a model with ordinary group-like symmetry. In 2+1d dimensions or higher, one can generalize the standard gauging procedure by stacking a symmetry-enriched topological order before gauging the symmetry. This generalized gauging procedure allows us to realize a large class of non-invertible symmetries. In this talk, I will describe the generalized gauging of finite group symmetries in 2+1d lattice models. This talk will be based on my ongoing work with L. Bhardwaj, S.-J. Huang, S. Schäfer-Nameki, and A. Tiwari.

In "Higher Operations in Perturbation Theory", Gaiotto, Kulp, and Wu discussed Feynman integrals that control certain deformations in quantum field theory. The corresponding integrands are differential forms in Schwinger parameters. Specifically, the integrand $\alpha$ is associated to a single topological direction of the theory.
I will show how the combinatorial properties of graph polynomials lead to a relatively simple, explicit formula for $\alpha$, that can be evaluated quickly with a computer. This is interesting for two reasons. Firstly, knowing the explicit formula leads to an elementary proof of the fact that $\alpha$ squares to zero, which asserts the absence of quantum corrections in topological field theories of two (or more) dimensions, known as Kontsevich's formality theorem. Secondly, the underlying constructions and proofs are not intrinsically limited to topological theories. In this sense, they serve as a particularly instructive example for simplifications that can occur in Feynman integrals with numerators.